Can anyone suggest a budget/moderate priced hotel near the Tel Aviv airport? We'll have one night in the city before moving to Jerusalem, preferably by public transportation. Any advice is appreciated.
The "Tel Aviv" airport is actually between Tel Aviv and Jerusalem; it's about 25 minutes to Tel Aviv or 50 minutes to Jerusalem, but both are equally convenient to reach from it. Therefore, there's no need to stay in Tel Aviv for one night on arrival and then move to Jerusalem; just go to right to Jerusalem.
Furthermore, according to the participants on Tripadvisor (the busiest Israel online forum), the only hotel right near the airport is totally inconvenient unless you're visiting people at the nearby hospital. It has no other services, nearby restaurants, or transportation into either Jerusalem or Tel Aviv. Whereas, hotels actually in these cities can be reached easily by taxi (both), bus (Jerusalem), sherut (Jerusalem - works very much like SuperShuttle in the US) or train (Tel Aviv).
If you're worried about timing of your arrival, the only issue is that on Shabbat (sundown Friday to sundown Saturday) or holidays, mass transit does not run. Therefore, you can't take the train from the airport to Tel Aviv or the bus to Jerusalem. However, the sherut to Jerusalem and taxis to both cities run as normal on Saturdays and holidays. For very early or very late, the bus to Jerusalem won't be running. Taxis and sheruts run any time the airport is open, and there is an hourly service all night on the airport to Tel Aviv train line (more frequent during the day, of course).
For getting between Tel Aviv and Jerusalem themselves (as opposed to the airport), the best way is by bus. The 405 goes from the Jerusalem Central Bus Station to the Tel Aviv Central Bus Station, and the 480 goes from the Jerusalem Central Bus Station to the Tel Aviv Arlozorov Station (closer to the center, so usually better). These buses leave every 10-15 minutes all day, and there's no need to reserve - just buy your ticket and get in line, and you get on the next available bus. Luggage goes under the bus, and it takes about an hour. I think the price is 16 shekels - less than 5 USD!
There is a train between Jerusalem and Tel Aviv that's supposed to be very scenic, but it takes almost twice as long, and arrives at the far outskirts of Jerusalem. They're building a high speed rail now. However, trains are definitely the way to go from Tel Aviv to Haifa to Akko; again, these have frequent service with no need for reservations, and are very inexpensive.
Finally, if you are looking for a hotel in Tel Aviv, I had a good stay at the Hotel Maxim. It is not right on the beach, so it's cheaper than the ones that are. The breakfast was fine, but not quite as spectacular as some other hotels seem to promise. And although I was warned my room would be small, as someone used to European hotel rooms, it was fine. It's near various restaurants and other services, as well as a block from Ben Yehuda Street with its very frequent buses to other parts of the city.
I'd certainly stay here again, and as I said it was cheaper than other recommended hotels (Israeli hotels are definitely not cheap compared with much of Europe). http://www.telavivmaxim.com/
Do look at the Tripadvisor Israel Forum for lots of other information: https://www.tripadvisor.com/ShowForum-g293977-i1733-Israel.html
Quite a long time ago we stayed at the Hotel Deborah in Tel Aviv. It is not far from the beach and seems still a decent place to stay for a night or two.